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The usual way of teaching is to explain the numbers that are element of the reals and naturals as being the same, this was a perfectly valid way of understanding for me, why do some consider '2' as an element of the reals, different to '2' as an element of the natural numbers due to their definition in set theory and why do we do this?

As an inexperienced student, is this worth taking note of? Is there any well defined consensus on whether an element x is the same as either a real of natural number? Is it better to simply consider '2' as the same abstract object.

We just refer to the number as one thing, 'the number 2' is there many things that can be referred to like this?

Here's an example of such an idea

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  • The definition of number in a philosphical sense (the essence of number) is a complex and yet unsolved issue. Mathematics has with different theories: arithmetic, set theory that define how numbers work. Aug 28, 2022 at 16:55
  • @MauroALLEGRANZA for most purposes, I shall think of a single abstract entity then.
    – user37577
    Aug 28, 2022 at 17:31
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    To first really understand and visualize the real number line, you need to address this simple paradox: it's a mundane fact from the denseness of the rational numbers in the real numbers that between every 2 distinct irrational numbers there's a rational number and symmetrically between every 2 distinct rational numbers there's an irrational number. Then how is it possible from our mundane intuition that Cantor rightly proved there must be much much more irrational numbers than all the rational numbers in terms of cardinality?... Aug 28, 2022 at 19:27
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    Modern competent mathematicians who are aware of the Benacerraf's identification problem do notice the faulty naive Platonism conception and more towards some kind of structualism such as reflected in the widespread use of category theory: arguing that the Platonic attempt to identify the "true" reduction of natural numbers to pure sets, as revealing the intrinsic properties of these abstract mathematical objects, is impossible....the relation of set theory to natural numbers cannot have an ontologically Platonic nature. Aug 28, 2022 at 21:51
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    In Peano arithmetic, for example, there are no reals, so 2 will be a natural number only. In ZFC, {∅,{∅}} represents 2 as both a natural and a real, reflecting the naive approach. One context where you would want to distinguish them is meta-theory, where you consider several formal systems and talk about relations between their models. For example, you may want to say that there is a map from a model of arithmetic that embeds it into a model of reals and 2-real is the image of 2-natural under this map. Once people say that they may further say that they "identify" naturals with their images.
    – Conifold
    Aug 29, 2022 at 8:51

2 Answers 2

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In a strict sense they are not the same, the naturals N and the reals R, by the usual construction, are different things. But, it happens that there is a natural embedding f: N---> R such that the set N (the naturals) and f(N) (a subset of R) are naturally isomorphic ("equivalent in a broad sense") and, thus,in a certain sense it is indifferent to speak of the properties of 2 in N or of the properties of f(2) in R.

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Kronecker said:

God created the integers; all else is the work of man.

This is why we call the integers the natural numbers because they seem to be a work of nature rather than than an invention of man. But if one follows this thought, then one see's that this cannot possibly be correct. If God created the integers then he also created all other numbers - the reals, the imaginaries, the Lie algebras, the finite groups and so on. All of them are as natural as the integers. That we do not think so is merely that we are unfamiliar with them.

This position, whether you believe in God or not, is called mathematical Platonism. In fact, its such a truncated version of Platonism that its serious mistake to think this is what Platonism is about. But we are stuck with the name. In this notion numbers have ideal existences that are only accessible to the inner sense of the intellect. Here, numbets are unique and there are no copies. Although this notion to contemporary ears sounds bizarre and few mathematicians would give it credence, it is more or less how most mathematicians naively go about working with them.

The other main option is nominalism. Here the number three is understood, for example, to be about the collection of ALL three objects. This option became popular with the advent of set theory which is a theory of collections. Still, even this approach can be suspect since the unrestricted use of ALL as Russell pointed out leads to inconsistencies. Also, personally speaking, I think it unwieldy and not at all how I think of numbers and I suspect, pretty much everybody else. Its rather like describing a book as a collection of pages, which even if true, is only partially true, and doesn't get to the idea of a book as a thing in itself.

If you ascribe real essences to numbers then its of course wrong to think of 2 as an integer as different from 2 as a real. That people do so may seem from this point of view completely wrongheaded. Nevertheless, one can argue that they do so only as a manner of speaking and are actually focusing on different properties of the same thing because those are the properties that they are interested in - either in the pure mathematics sense where numbers are their own end - or practically because what they are modelling requires only those properties.

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